I'll kick it off with the obvious one: LP requires linearity; the objective function must be linear, the constraint equations must be linear, each decision variable is multiplied by a constant coefficient with no multiplying between decision variables. Obviously, no higher order terms, no logs, etcs. So boo hoo, what does this force us into?
ASSUMPTIONS: Proportionality- change in variable results in a proportionate change in that variables contribution to the value of the function.
We must also assume that the function value is the sum of the contributions of each term. Decision variables can be devided into messy fractional values.
Most of all, LP assumes certainty; the coefficients are known and constant.
Sure, computers can do easilly do their part in LP. The challenge is in the the formulation of the problem. How do you translate some statement or problem into a system of linear equations?
Then there is always the fight between tightening constraints and loosening constraints. You want optimal solution, and LP is good and easy once you have applied it. But often time it will get you a result or solution... but it won't be what you want in a situation.
Was this too vague?